We propose a method for tracing implicit real algebraic curves defined by polynomials with rank-deficient Jacobians. For a given curve $f^{-1}(0)$, it first utilizes a regularization technique to compute at least one witness point per connected component of the curve. We improve this step by establishing a sufficient condition for testing the emptiness of $f^{-1}(0)$. We also analyze the convergence rate and carry out an error analysis for refining the witness points. The witness points are obtained by computing the minimum distance of a random point to a smooth manifold embedding the curve while at the same time penalizing the residual of $f$ at the local minima. To trace the curve starting from these witness points, we prove that if one drags the random point along a trajectory inside a tubular neighborhood of the embedded manifold of the curve, the projection of the trajectory on the manifold is unique and can be computed by numerical continuation. We then show how to choose such a trajectory to approximate the curve by computing eigenvectors of certain matrices. Effectiveness of the method is illustrated by examples.
翻译:我们提出一种方法来追踪由低级雅各布人多元体界定的隐性实际代数曲线。 对于给定的曲线 $f ⁇ -1}(0) 美元, 它首先使用正规化技术来计算曲线每个连接部分的至少一个证人点。 我们改进这一步骤, 为测试 $ff ⁇ -1}(0) 美元 的空虚建立充分的条件 。 我们还分析汇合率, 并对改进证人点进行错误分析 。 证人点是通过计算随机点的最小距离, 以利滑的柱嵌入曲线, 同时对本地微型曲线的剩余美元进行处罚 。 为了从这些证人点开始追踪曲线的曲线, 我们证明, 如果有人沿着曲线嵌入的柱形形形的圆形周围的轨迹拖动随机点, 则对柱形轨迹的预测是独一无二的, 并且可以用数字的连续性来计算。 我们然后通过计算某些矩阵的外观来显示如何选择这样的轨线以接近曲线的曲线。 方法的有效性通过示例来说明。